This is part of the satirical nature of the novel.
Flatland is well-known as a satirical novel. The satire lies partly in its commentary on the social hierarchy of Victorian Britain (as denoted by the different types of polygon in Flatland), but also in the way the society is unable to accept new ideas. Even when they see others refusing to accept their new ideas, they lack the introspection necessary to notice the same blindness in themselves.
I've read (but can't find a source for this now) that in around 1900 the French Academy of Sciences declared that there were no more important inventions to be made, that science had more or less reached its pinnacle. They would have laughed at the idea of anyone saying that in 1800, but they failed to see that it was just as ridiculous to say it in 1900, or at any other time.
Scientists can ridicule theories of the past which have been disproved, without being able to accept that their own modern theories could be disproved just as easily. It's easy for physicists of today to laugh at the idea of invisible aether permeating reality, or of heat as a substance which flows, but less easy for them to contemplate the idea that their own modern models of reality, such as string theory, could be equally wrong.
Atheists can scoff at religious folk for believing God exists without proof of his existence, while they themselves believe God doesn't exist without proof of his non-existence. They can see the illogicality in others' "faith" that God exists, but - despite the parallelism - they fail to see the illogicality in their own "faith" that God doesn't exist.
Nationalists, supporters of sports teams, proponents of "Mac vs PC" or "Star Wars vs Star Trek" or "vim vs emacs" - many people who are in one of those groups but not another might again fail to spot the parallels. People might criticise rabid sports fans for their irrational hatred of opposing teams, without realising that the same applies to their own rabid nationalism and irrational hatred of opposing countries.
It's a common theme in politics nowadays to turn arguments against one party around to attack another party in the same way. In US terms, Democrats might criticise the Republican party for doing XYZ and having leaders who do ABC, without accepting that the Democrat party and its leaders have also done very similar things.
Hypocrisy and failure to use analogy for introspection are common themes in society, in Edwin Abbott's day and even still today. It's not at all strange or unrealistic for A. Square to fail to see that the same arguments he applied to the Linelanders can also be applied to the Flatlanders. People find it easy to see the world they know, but even having tried to convince more ignorant people of its existence, they have trouble accepting that they themselves are more ignorant than others and that there's more to the world than they can see too. "Those arguments don't apply to me! The Linelanders were wrong because there is more than one dimension, but there aren't more than two dimensions!"
It's true that in the real world the analogies are usually less stark and clear. Perhaps not all of the arguments used by proponents of A criticising B also apply exactly to A as well. This is why even intelligent people can sometimes fail to see how their own arguments can be used against them. But by using a caricaturised, unrealistically clear version, Abbott is making a more general point. This is a common technique used in satire in general: don't portray the world exactly as it is to make your points about it, but instead build a simplified version in which those points come out more clearly. The transitions from 1 to 2 dimensions and from 2 to 3 dimensions are much more similar to each other than many of the other comparisons I've discussed above, so it seems even more ridiculous that A. Square fails to grasp that similarity. But this is only an exaggerated version of the same ridiculousness of people failing to grasp other, less close, similarities in the real world.